# Upper bound for $-t \log t$

While reading Csiszár & Körner's "Information Theory: Coding Theorems for Discrete Memoryless Systems", I came across the following argument:

Since $f(t) \triangleq -t\log t$ is concave and $f(0) = 0$ and $f(1) = 0$, we have for every $0 \leq t \leq 1-\tau$, $0 \leq \tau \leq 1/2$, $$|f(t) - f(t+\tau)| \leq \max (f(\tau), f(1-\tau)) = -\tau \log\tau$$

I can't make any progress in seeing how this bound follows from the properties of $f(t)$. Any insights would be greatly appreciated.

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I would say, $\log 0$ is not defined, thus $f(0)$ is undefined, too. They might think about its continuous continuation (is this really a word?), though. – Paŭlo Ebermann Nov 7 '11 at 1:55

The function $g$ defined on the interval $I=[0,1-\tau]$ by $g(t)=f(t)-f(t+\tau)$ has derivative $g'(t)=-\log(t)+\log(t+\tau)$. This derivative is positive hence $g$ is increasing on $I$ from $g(0)=-f(\tau)<0$ to $g(1-\tau)=f(1-\tau)>0$. For every $t$ in $I$, $g(t)$ belongs to the interval $[-f(\tau),f(1-\tau)]$, in particular $|g(t)|\leqslant\max\{f(\tau),f(1-\tau)\}$.